From left, Sens. John Barrasso, R-Wyo., and John Thune, R-S.D., Senate Majority Leader Mitch McConnell, R-Ky., and Senate Majority Whip John Cornyn, R-Texas, conduct a news conference in the Capitol on Wednesday after the policy lunches. (Tom Williams/CQ Roll Call)

Senate Republicans say they are willing to join their House counterparts in a postelection fight over border wall funding but recognize that their chamber will be more constrained by the need for Democratic votes.